Non-Hermitian ultra-strong bosonic clustering through interaction-induced caging

TL;DR

The paper addresses how non-Hermitian pumping, bosonic interactions, and band topology can produce ultra-strong bosonic condensation beyond simple NHSE localization. Using a minimal 1D non-Hermitian SSH lattice with asymmetric hoppings and a local density nonlinearity $U n_{x_0}^2$, the authors study two-boson dynamics via Schrödinger evolution and monitor observables such as the two-boson density $\rho_x(t)$ and clustering indicators. They uncover an interaction-induced caging mechanism, organized in the two-particle configuration space, that synergizes with topological band hybridization to yield boundary clustering far exceeding expectations and persisting to larger boson numbers. The results open new routes to control bosons in non-Hermitian many-body systems and suggest experimental realizations in photonic lattices, optical platforms, and circuit-QED architectures.

Abstract

We uncover a new mechanism whereby the triple interplay of non-Hermitian pumping, bosonic interactions and nontrivial band topology leads to ultra-strong bosonic condensation. The extent of condensation goes beyond what is naively expected from the interaction-induced trapping of non-Hermitian pumped states, and is based on an emergent caging mechanism that can be further enhanced by topological boundary modes. Beyond our minimal model with 2 bosons, this caging remains applicable for generic many-boson systems subject to a broad range of density interactions and non-Hermitian hopping asymmetry. Our novel new mechanism for particle localization and condensation would inspire fundamental shifts in our comprehension of many-body non-Hermitian dynamics and opens new avenues for controlling and manipulating bosons.

Figures (9)

• Figure 1: Two-boson correlation dynamics and associated spatial density distributions in a boundary-interacting system. The dramatic enhancement of bosonic clustering at site $x=1$ is captured by $P_{11}$ [Eq. \ref{['eq:Pmn']}], which measures the probability of finding both bosons at this site. This correlation is significantly stronger (red curves in panel e) when both the non-Hermitian skin effect (NHSE, $r>0$) and boundary interaction ($U>0$) are present, demonstrating their synergistic effect on boson trapping. This enhanced clustering is studied for different initial states $(1,1)$, $(4,4)$, and $(1,4)$ evolving under $H$ [Eq. \ref{['eq:H']}]. Supporting this observation, panels (a-d) show the spatial density evolution $\rho_x(t)$ [Eq. \ref{['eq:bosonnum']}] over time $t \in [0, 10]$, with corresponding time-averaged profiles $\bar{\rho}_i$ [Eq. \ref{['eq:timeaverage_rho']}] plotted above. The density distributions contrast four scenarios: Hermitian ($r=1$) without (a) and with (b) interaction, and non-Hermitian ($r=4$) without (c) and with (d) interaction. Notably, the interaction term $U$, despite being typically repulsive, acts as an effective trap at $x_0=1$ during non-equilibrium evolution when combined with leftward NHSE, as evidenced by the bright regions in the heatmaps. Parameters: $t_0=3$, and for $r=4$, $t_L=1.6$ and $t_R=0.4$.
• Figure 2: Extent of two-boson clustering due to a boundary density interaction at $x_0=1$ [Eq. \ref{['eq:H']}], and its correspondence with the boundary clustering of 2-boson spectral bands. (a) Time-averaged two-boson clustering probability $\bar{P}_{11}$ [Eq. \ref{['eq:smoothedPmn']}] at site $x=1$, in the parameter space of non-Hermitian hopping asymmetry $r$ and density interaction strength $U$. (b) Corresponding two-boson spectra (purely real) at $r=4$ ($t_L=1.6$ and $t_R=0.4), t_0=3$, which features five bands, with bands 2 and 4 being containing a pair of interaction-hybridized bulk and topological bosons. As $U$ increases, it creates a group of eigenstates at $E\approx U$ that exhibits strong clustering $\psi_{11}$ (red) at $x=x_0=1$. Strong $\psi_{11}$ clustering in the hybrid topological band 4 leads to suppressed $\bar{P}_{11}$ for the boundary-localized initial state $(1,1)$ (circled in dashed black), but enhanced $\bar{P}_{11}$ clustering for the initial state $(1,4)$ (solid black). By contrast, strong $\psi_{11}$ clustering in the bulk bands 3 and 5 (circled in light blue) corresponds to enhanced $\bar{P}_{11}$ for the bulk initial state $(4,4)$.
• Figure 3: Ultrastrong boundary clustering $\bar{P}_{11}$ from interaction-induced caging, for initial state (2,2) within a $x_0=3$ cage for a 5 unit cell chain. (a) Snapshots of the time-smoothed ($\Delta t=2$) two-boson correlation probability $\bar{P}_{11}$ [Eq. \ref{['eq:smoothedPmn']}] for (a1) non-interacting and (a2) interacting cases. The L-shaped cage in (a2) traps the bosons tightly. (b1) Evolution of probability of both bosons at the left boundary, with greatly enhanced $\bar{P}_{11}(t)$ only for the non-Hermitian interacting case (red). (b2) When only one boson is present, the boundary density $\bar{\rho}_1(t)$ remains low regardless of whether a $U$ barrier is present, showcasing that the caging mechanism is an interaction effect.
• Figure 4: Persistence of ultrastrong boundary clustering $\bar{P}_{11}$ from interaction-induced caging, even for initial state (4,4) outside a $x_0=3$ cage. (a) Snapshots of the time-smoothed ($\Delta t=2$) two-boson correlation probability $\bar{P}_{11}$ (Eq. \ref{['eq:smoothedPmn']}) for (a1) non-interacting and (a2) interacting cases. The bosons still gradually penetrate the L-shaped cage due to the NHSE towards (1,1), and remain trapped in it after that ($t=3,8$). (b1) Eventually, $\bar{P}_{11}(t)$ is ultra-enhanced only for the non-Hermitian interacting case (red), with (b2) negligible enhancement of boundary localization by $U$ in the single-boson case, similar to (b1,b2) of Fig. \ref{['fig:initial22x03']}. (c) Time-averaged two-boson probability $\bar{P}_\text{cage}$ within the $x_0\times x_0$ cage as a function of hopping asymmetry $r$. For initial states $(2,2)$ and $(4,4)$ with coincident bosons, the $U=1$ cases exhibit much larger $\bar{P}_\text{cage}$ due to interaction-induced caging. But for $(2,4)$, non-coincident initial bosons do not interact appreciably and the effect of interactions is negligible.
• Figure 5: Single-boson spectrum of the non-Hermitian SSH model [Eq. \ref{['smeq:H']} with $U=0$] under open boundary conditions with $t_L=1.6$, $t_R=0.4$ and $t_0=3$. (a) The energy spectrum shows two bulk bands ($E_1$ and $E_2$) separated by topological zero modes ($E_0$). (b) Localization profile $|\psi|^2$ of left boundary-localized states in the bulk bands. (c) Localization of the topological zero mode, is also confined at the left boundary.